A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

TIIEORY OF ISOGONAL AND ISOTOMIC POINTS. 167

DEF. — The points X, Y are called isogonal conjugates
with respect to the triangle ABC.
COR. 1. —If X, Y be isogonal conjugates with respect
to a triangle, the three rectangles contained by the distances of X, Y from the sides of the triangle are equal to
one another.
CoR. 2.-The middle point of the line XY is equally
distant from the projections of the points X, Y on the
three sides of the triangle.
Exercises.
1. The sum of the angles BXC, BYC is 180~+ A.
2. The line joining any two points, and the line joining their
isogonal conjugates, with respect to a triangle, subtend at any
vertex of the triangle angles which are either equal or supplemental.
3. AM2: AN2:: BM. MC: BN. NC. (STEINER.)
4. If the lines AX, AY meet the circumcircle of the triangle
ABC in M', N', then the rectangles AB. AC, AM. AN', and
AM'. AN are equal to one another.
5. The isogonal conjugate of the point M' is the point at infinity on the line AN'.
6. If three lines through the vertices of a triangle meet the
opposite sides in collinear points, their isogonal conjugates will
also meet them in collinear points.
7. If upon the sides of a triangle ABC three equilateral triangles ABC', BCA', CAB' be described either externally or internally, the isogonal conjugate of the centre of perspective
of the triangles ABC, A'B'C', is a point common to the three
Apollonian circles of ABC. (See Cor. 3, p. 86.)
8. If the lines MX, QY in fig. Prop. 1, intersect in D, and
the lines MP, NQ inE, the lines AD, AE are isogonal conjugates
with respect to the angle BAC.
9. If D, E be the points where two isogonal conjugates, with
respect to the angle BAC, meet the base BC of the triangle BAC
and if perpendiculars to AB, AC at the points B, C meet the per
pendiculars to BC at D, E in the points D', E'; D", E", respee
tively; then BD'. BE': CD". CE":: AB4: AC4.
10. In the same case BD. BE: CD. CE:; AB2:AC2.
N 2